For the Kuramoto oscillators with small inertia, we present several quantitative estimates on the relaxation dynamics and formational structure of a phase-locked state (PLS) for some classes of initial configurations. In a super-critical regime where the coupling strength is strictly larger than the diameter of natural frequencies, we present quantitative relaxation dynamics on the collision numbers and the structure of PLS. In a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies, we provide a sufficient condition for an asymptotically PLS solution. In particular, we show the existence of slow relaxation to a PLS, when there are exactly two natural frequencies. This generalizes the earlier results of Choi et al. ["Asymptotic formation and orbital stability of phase locked states for the Kuramoto model," Physica D241, 735-754 (2012)10.1016/j.physd.2011.11.011; Choi et al. "Complete synchronization of Kuramoto oscillators with finite inertia," Physica D240, 32-44 (2011)]10.1016/j.physd.2010.08.004.
Bibliographical noteFunding Information:
The work of Y.-P. Choi is partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A6A3A03039496). The work of S.-Y. Ha is partially supported by NRF-2011-0015388, and the work of S. E. Noh is supported by 2012 Research Fund of Myongji University.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics